a shorter proof: Martin’s axiom and the continuum hypothesis

This is another, shorter, proof for the fact that M⁢Aℵ0 always holds.

Let (P,≤) be a partially ordered set and 𝒟 be a collection of subsets of P. We remember that a filter G on (P,≤) is 𝒟-generic if G∩D≠∅ for all D∈𝒟 which are dense in(P,≤). (In this context “dense” means: If D is dense in (P,≤), then for every p∈P there’s a d∈D such that d≤p.)

Let (P,≤) be a partially ordered set and 𝒟 a countable collection of dense subsets of P. Then there exists a 𝒟-generic filter G on P. Moreover, it could be shown that for every p∈P there’s such a 𝒟-generic filter G with p∈G.

Proof.

Let D1,…,Dn,… be the dense subsets in 𝒟. Furthermore let p0=p. Now we can choose for every 1≤n<ω an elementpn∈P such that pn≤pn-1 and pn∈Dn. If we now consider the set G:={q∈P∣∃n<ω⁢ s.t. ⁢pn≤q}, then it is easy to check that G is a 𝒟-generic filter on P and p∈G obviously. This completes the proof.
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